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Two different Hamiltonian formulations of the metric gravity are discussedand applied to describe a free gravitational field in the $d$ dimensionalRiemann space-time. Theory of canonical transformations, which relateequivalent Hamiltonian formulations of the metric gravity, is investigated indetails. In particular, we have formulated the conditions of canonicity fortransformation between the two sets of dynamical variables used in ourHamiltonian formulations of the metric gravity. Such conditions include theordinary condition of canonicity known in classical Hamilton mechanics, i.e.,the exact coincidence of the Poisson (or Laplace) brackets which are determinedfor the both new and old dynamical Hamiltonian variables. However, in additionto this any true canonical transformations defined in the metric gravity, whichis a constrained dynamical system, must also guarantee the exact conservationof the total Hamiltonians $H_t$ (in the both formulations) and preservation ofthe algebra of first-class constraints. We show that Dirac's modifications ofthe classical Hamilton method contain a number of crucial advantages, whichprovide an obvious superiority of this method in order to develop variousnon-contradictory Hamiltonian theories of many physical fields, when a numberof gauge conditions are also important. Theory of integral invariants and itsapplications to the Hamiltonian metric gravity are also discussed. ForHamiltonian dynamical systems with first-class constraints this theory leads toa number of peculiarities some of which have been investigated.

Metric gravity in the Hamiltonian form. Canonical transformations.  Dirac's modifications of the Hamilton method and integral invariants of the  metric gravity